Question

In a factory, the expected number of accidents per day is linearly related to the overtime hours \(x\). On a day with \(x = 1000\) overtime hours there were 8 accidents; on a day with \(x = 400\) hours there were 5 accidents. What is the expected number of accidents when no overtime is logged (\(x = 0\))?

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

Model accidents as a straight line \(A = ax + c\) in overtime hours \(x\); \(c\) is exactly the value at \(x=0\) the question wants.

Answer 1

The Correct Option is B

Step 1: Set up the linear model.
The problem says the expected number of accidents per day is linearly related to the overtime hours \(x\). A linear relation between two quantities always has the form \(A = ax + c\), where \(a\) is the rate of change (extra accidents per extra overtime hour) and \(c\) is the value of \(A\) when \(x = 0\), which is exactly what the question is asking for.

Step 2: Write the two given conditions as equations.
When \(x = 1000\), \(A = 8\), so
\[ 1000a + c = 8 \quad \text{...(1)} \]
When \(x = 400\), \(A = 5\), so
\[ 400a + c = 5 \quad \text{...(2)} \]

Step 3: Eliminate \(c\) and solve for \(a\).
Subtract equation (2) from equation (1):
\[ (1000a + c) - (400a + c) = 8 - 5 \]
\[ 600a = 3 \]
\[ a = \frac{3}{600} = \frac{1}{200} \]

Step 4: Solve for \(c\).
Put \(a = \frac{1}{200}\) back into equation (2):
\[ 400 \times \frac{1}{200} + c = 5 \]
\[ 2 + c = 5 \]
\[ c = 3 \]

Step 5: Read off the answer.
Since \(c\) is the value of \(A\) at \(x = 0\), the expected number of accidents with no overtime logged is 3, which is option (B).
\[ \boxed{3} \]